A theorem on fractional ID -(g,f)(g,f)- factor-critical graphs

Let $a$, $b$ and $r$ be three nonnegative integers with $2\leq a\leq b-r$, let $G$ be a graph of order $p$ satisfying the inequality $p(a+r) \geq (a+b-3)(2a+b+r)+1$, and let $g$ and $f$ be two integer-valued functions defined on $V(G)$ satisfying $a\leq g(x)\leq f(x)-r\leq b-r$ for every $x\in V(G)$. A graph $G$ is said to be fractional ID-$(g,f)$-factor-critical if $G-I$ contains a fractional $(g,f)$-factor for every independent set $I$ of $G$. In this paper, we prove that $G$ is fractional ID-$(g,f)$-factor-critical if \operatorname{bind}(G)((a+r)p - (a+b-2)) > (2a+b+r-1)(p-1), which is a generalization of a previous result of Zhou

Spectral radius and k-factor-critical graphs

For a nonnegative integer $k$, a graph $G$ is said to be $k$-factor-critical if $G-Q$ admits a perfect matching for any $Q\subseteq V(G)$ with $|Q|=k$. In this article, we prove spectral radius conditions for the existence of $k$-factor-critical graphs. Our result generalises one previous result on perfect matchings of graphs. Furthermore, we claim that the bounds on spectral radius in Theorem 3.1 are sharp.Comment: 12 page

Network-Structured BST/MBO Composites Made from Core-Shell-Structured Granulates

A finite element method (FEM)-based simulation approach to predict the tunability in composite materials was developed and tested with analytical data. These tests showed good prediction capabilities of the simulation for the test data. The simulation model was then used to predict the tunability of a network-structured composite, where the dielectric phase formed clusters in a paraelectric network. This was achieved by simulating a reciprocal core-shell unit cell of said network. The simulation showed a high tunability for this network model, exceeding the tunability of the analytically evaluated layered, columnar, and particulate model. The simulation results were experimentally verified with a Ba0.6Sr0.4TiO3/Mg3B2O6 (BST/MBO) composite, where core-shell granulates were made with a two-step granulation process. These structured samples showed higher tunability and dielectric loss than the unstructured samples made for comparison. Overall, the structured samples showed higher tunability to loss ratios, indicating their potential for use in tunable radio frequency applications, since they may combine high performance with little energy loss

Network-Structured BST/MBO Composites Made from Core-Shell-Structured Granulates

A finite element method (FEM)-based simulation approach to predict the tunability in composite materials was developed and tested with analytical data. These tests showed good prediction capabilities of the simulation for the test data. The simulation model was then used to predict the tunability of a network-structured composite, where the dielectric phase formed clusters in a paraelectric network. This was achieved by simulating a reciprocal core-shell unit cell of said network. The simulation showed a high tunability for this network model, exceeding the tunability of the analytically evaluated layered, columnar, and particulate model. The simulation results were experimentally verified with a Ba₀.₆Sr₀.₄TiO₃/Mg₃B₂O₆ (BST/MBO) composite, where core-shell granulates were made with a two-step granulation process. These structured samples showed higher tunability and dielectric loss than the unstructured samples made for comparison. Overall, the structured samples showed higher tunability to loss ratios, indicating their potential for use in tunable radio frequency applications, since they may combine high performance with little energy loss

A neighborhood condition for graphs to have restricted fractional (g,f)-factors

Let $h$ be a function defined on $E(G)$ with $h(e)\in[0,1]$ for any $e\in E(G)$. Set $d_G^{h}(x)=\sum_{e\ni x}h(e)$. If $g(x)\leq d_G^{h}(x)\leq f(x)$ for every $x\in V(G)$, then we call the graph $F_h$ with vertex set $V(G)$ and edge set $E_h$ a fractional $(g,f)$-factor of $G$ with indicator function $h$, where E_h=\{e:e\in E(G),h(e)>0\}. Let $M$ and $N$ be two sets of independent edges of $G$ with $M\cap N=\emptyset$, $|M|=m$ and $|N|=n$. If $G$ admits a fractional $(g,f)$-factor $F_h$ such that $h(e)=1$ for any $e\in M$ and $h(e)=0$ for any $e\in N$, then we say that $G$ has a fractional $(g,f)$-factor with the property $E(m,n)$. In this paper, we present a neighborhood condition for the existence of a fractional $(g,f)$-factor with the property $E(1,n)$ in a graph. Furthermore, it is shown that the neighborhood condition is sharp

A Neighborhood Condition for Fractional ID-[A, B]-Factor-Critical Graphs

Let $G$ be a graph of order $n$, and let $a$ and $b$ be two integers with $1 \le a \le b$. Let $h : E(G) \rightarrow [0, 1]$ be a function. If $a \le \Sigma_{ e \ni x } h(e) \le b$ holds for any $x \in V (G)$, then we call $G[F_h]$ a fractional $[a, b]$-factor of $G$ with indicator function $h$, where F_h = \{ e \in E(G) : h(e) > 0 \} . A graph $G$ is fractional independent-set-deletable $[a, b]$-factor-critical (in short, fractional ID-$[a, b]$-factor-critical) if $G − I$ has a fractional $[a, b]$-factor for every independent set $I$ of $G$. In this paper, it is proved that if $n \ge \frac{(a+2b)(2a+2b-3)+1}{b}$, $\delta (G) \ge \frac{bn}{a+2b} + a$ and $| N_G(x) \cup N_G(y) | \ge \frac{(a+b)n}{a+2b}$ for any two nonadjacent vertices $x, y \in V (G)$, then $G$ is fractional ID-$[a, b]$-factor-critical. Furthermore, it is shown that this result is best possible in some sense